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Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Given the coordinate matrix of \( x \) relative to a (nonstandard) basis \( B \) for \(\mathbb{R}^n\), find the coordinate matrix of \( x \) relative to the standard basis.
\[ B = \{(3, -1), (0, 1)\} \]
\[
[x]_B =
\begin{bmatrix}
3 \\
5
\end{bmatrix}
\]
\[ [x]_S =
\begin{bmatrix}
\hspace{1cm} \\
\hspace{1cm}
\end{bmatrix}
\]
Explanation:
- The problem involves converting the coordinate matrix of a vector \( x \) from a given nonstandard basis \( B \) to the standard basis.
- The basis \( B \) consists of the vectors \( (3, -1) \) and \( (0, 1) \).
- The coordinate matrix of \( x \) relative to \( B \) is \(\begin{bmatrix} 3 \\ 5 \end{bmatrix}\).
- The expression \([x]_S\) represents the coordinate matrix of \( x \) relative to the standard basis, which needs to be determined.](https://content.bartleby.com/qna-images/question/0fc3766d-bc8c-44dc-b95b-e319f0e06bc0/aa7a6ec2-3906-4785-b124-94d755aa2a77/ldvquw8_thumbnail.jpeg)
Transcribed Image Text:Given the coordinate matrix of \( x \) relative to a (nonstandard) basis \( B \) for \(\mathbb{R}^n\), find the coordinate matrix of \( x \) relative to the standard basis.
\[ B = \{(3, -1), (0, 1)\} \]
\[
[x]_B =
\begin{bmatrix}
3 \\
5
\end{bmatrix}
\]
\[ [x]_S =
\begin{bmatrix}
\hspace{1cm} \\
\hspace{1cm}
\end{bmatrix}
\]
Explanation:
- The problem involves converting the coordinate matrix of a vector \( x \) from a given nonstandard basis \( B \) to the standard basis.
- The basis \( B \) consists of the vectors \( (3, -1) \) and \( (0, 1) \).
- The coordinate matrix of \( x \) relative to \( B \) is \(\begin{bmatrix} 3 \\ 5 \end{bmatrix}\).
- The expression \([x]_S\) represents the coordinate matrix of \( x \) relative to the standard basis, which needs to be determined.
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